The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [S2] in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded d-dimensional cobordism category for all d. For d = 2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one, presented in [MW].
S. Galatius was partially supported by NSF grant DMS-0505740 and the Clay Institute.
M. Weiss was partially supported by the Royal Society and the Mittag-Leffler Institute.
"The homotopy type of the cobordism category." Acta Math. 202 (2) 195 - 239, 2009. https://doi.org/10.1007/s11511-009-0036-9